Subset Relation is Transitive

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The subset relation is transitive:

$\paren {R \subseteq S} \land \paren {S \subseteq T} \implies R \subseteq T$

Proof 1

\(\ds \) \(\) \(\ds \paren {R \subseteq S} \land \paren {S \subseteq T}\)
\(\ds \) \(\leadsto\) \(\ds \paren {x \in R \implies x \in S} \land \paren {x \in S \implies x \in T}\) Definition of Subset
\(\ds \) \(\leadsto\) \(\ds \paren {x \in R \implies x \in T}\) Hypothetical Syllogism
\(\ds \) \(\leadsto\) \(\ds R \subseteq T\) Definition of Subset


Proof 2

Let $V$ be a basic universe.

By definition of basic universe, $R$, $S$ and $T$ are all elements of $V$.

By the Axiom of Transitivity, $R$, $S$ and $T$ are all classes.

We are given that $R \subseteq S$ and $S \subseteq T$.

Hence by Subclass of Subclass is Subclass, $R$ is a subclass of $T$.

By Subclass of Set is Set, it follows that $R$ is a subset of $T$.

Hence the result.